Morita Bicategories of Algebras and Duality Involutions

TL;DR

This paper constructs and analyzes duality involutions within the Morita bicategory of finite-dimensional algebras, establishing a strict duality structure for the representation pseudofunctor and generalizing to tensor categories.

Contribution

It introduces a weak duality involution on the Morita bicategory of algebras and shows how the representation functor can be strictified as a duality pseudofunctor, extending to tensor categories.

Findings

Constructed a weak duality involution on the Morita bicategory of finite-dimensional algebras.

Established that the representation functor can be strictified as a duality pseudofunctor.

Generalized duality involutions to algebras in semisimple symmetric finite tensor categories.

Abstract

The notion of a weak duality involution on a bicategory was recently introduced by Shulman in [arXiv:1606.05058]. We construct a weak duality involution on the fully dualisable part of $\text{Alg}$, the Morita bicategory of finite-dimensional k-algebras. The 2-category $\text{KV}$ of Kapranov-Voevodsky k-vector spaces may be equipped with a canonical strict duality involution. We show that the pseudofunctor $\text{Rep}: \text{Alg}^{fd} \to \text{KV}$ sending an algebra to its category of finite-dimensional modules may be canonically equipped with the structure of a duality pseudofunctor. Thus $\text{Rep}$ is a strictification in the sense of Shulman's strictification theorem for bicategories with a weak duality involution. Finally, we present a general setting for duality involutions on the Morita bicategory of algebras in a semisimple symmetric finite tensor category.